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This work deals with the aggregation diffusion equation \[\partial_t \rho =\Delta_p\rho + \lambda div((K_a*\rho)\rho),\] where $K_a(x)=\frac{x}{|x|^a}$ isan attraction kernel and $\Delta_p$ is the so called $p$-Laplacian. We showthat the domain $a < p(d+1)-2d$ is subcritical with respect to the competitionbetween the aggregation and diffusion by proving that there is existenceunconditionally with respect to the mass. In the critical case we showexistence of solution in a small mass regime for an $L\ln L$ initial condition.

p-Laplacian Keller-Segel Equation: Fair Competition and Diffusion Dominated Cases